Guidelines for Papers to Be Presented at the ABAQUS Users' Conference

Multiscale Modeling of Polymer Composites: From Atomistic Simulation to Structural Analysis Takaya Kobayashi, Kensuke Ogawa Mechanical Design & Analysis Corporation Satoru Yamamoto, Riichi Kuwahara Dassault Systemes BIOVIA K.K. Ryosuke Matsuzaki Tokyo University of Science Abstract: A multiscale modeling procedure was presented to design thermosetting resins and products of polymer composites. Heat curing and cross-linking reaction steps were considered with activation energy and heat generation via molecular dynamics simulation of BIOVIA Materials Studio. The mechanical properties and adhesive strength for fillers were also estimated, and were assigned to SIMULIA Abaqus. Using the Abaqus fracture analysis capability, the crack propagation behavior, including matrix-failure and interface-failure, was investigated. Keywords: Polymer Composites, Multiscale Modeling, Atomistic Simulation and Structural Analysis. 1. Introduction Polymer composites formed by thermosetting resins and carbon fibers are promising materials in transportation systems, such as airplanes and automobiles, owing to their light weight and high stiffness. In thermosetting resins, combinations of epoxy and amine compounds are widely used as the base resin and curing agent, respectively. The curing process of the thermosetting resin was investigated, as shown in Figure 1. The mechanical behavior of the curing process can be characterized by thermo-rheological behaviors (typically, the expression of elasticity owing to gelling and the volumetric change owing to curing shrinkage). From an experimental point of view, the expression of elasticity is evaluated by viscoelastic measurements using a shear rheometer with parallel plates, whereas the curing shrinkage is measured typically by volumetric dilatometry. These measurements must be followed by an estimation of the curing (hardening) degree, which indicates the degree of progression of the curing reaction of the polymeric materials. For this purpose, differential scanning calorimeter (DSC) measurements provide the degree of curing, and the Kamal differential equation is a typical application to describe the progress of the curing reaction of the 1 thermosetting resin. Abaqus allows introduction of the Kamal model as a user-defined state variable, which makes it possible to provide the curing degree dependencies on the elastic modulus and the curing shrinkage. There are many varieties of epoxy and amine compounds, and their choice is a key strategy for material design because their combinations as well as curing conditions strongly affect the mechanical and thermal properties. The conventional trial-and-error procedures are expensive and time-consuming, and therefore, computer modeling is desired in order to reduce cost and accelerate the development. With this background, we propose a multiscale modeling procedure (Figure 2) to design an epoxy resin and a polymer composite. Our approach consists of two parts: an atomistic simulation for curing reaction and estimation of the mechanical and thermal properties, and an FEM simulation for the molding process and structure. In the first part, a heat curing reaction is simulated by the molecular dynamics (MD) method in order to explore how combinations of the base resin and curing agent affect the cross-linked structure and material properties. Mechanical and thermal properties of the epoxy resin and the adhesive strength between the resin and fillers are also predicted via MD simulation. In the FEM part, first, the distribution of the conversion of the curing reaction throughout the product is simulated. By assigning the obtained material properties to each part of the product in reference to the conversion of curing reaction, the stress and crack propagation are investigated for a part of the product and the whole product of the polymer composite. These approaches are expected to provide the prospect of solving the material irregularity in the actual polymer composite as shown in Figure 3 (Yoshimura, 2016). Figure 1. Experimental approaches to curing process of thermosetting resin. 2 Figure 2. Multiscale modeling of polymer composites. Figure 3. Irregularities in polymer composite. 2. Atomistic simulation The reaction of epoxy compounds with primary amines takes place through the following two steps: (1) 3 (2) When the epoxy group of the base resin approaches the amino group of the curing agent, a cross- link is constructed and a secondary amine is produced. The secondary amine reacts again with the epoxy compounds and produces a ternary amine. In this paper, we choose two epoxy compounds, DGEBA (Diglycidyl ether of bisphenol A) and TGDDM (Tetraglycidyl diaminodiphenylmethane). The number of epoxy groups is different in DGEBA and TGDDM, i.e., 2 and 4, respectively. For the amine compound, we choose 44DDS (4’4-Diaminodiphenylsulphone). The following calculations are performed by using the software package Materials Studio 2017 R2 (Dassault Systèmes BIOVIA). The Amorphous Cell module is used to construct atomistic models for two systems in stoichiometry; (a) 40 DGEBA and 20 44DDS, and (b) 40 TGDDM and 40 44DDS. The total number of atoms in the system is (a) 2,540 and (b) 3,600. The curing simulation is performed with the Forcite module and the COMPASSII force fields according to the molecular dynamics technique proposed by Okabe et al. (Okabe, 2016). After relaxation of the system at NPT ensemble, MD simulation is carried out to reproduce the cross- linked structure of the epoxy resin. When the epoxy group approaches to the amino group within the reaction range, the reaction probability k, which consists of the acceleration factor A and the activation energy G in Equation 3, is compared with a random number P (0-1) 4 ∆퐺 k= 퐴 푒푥푝 (− ) (3) 푅푇 where R is the gas constant and T is the local temperature. In the acceptable condition, the curing reaction occurs, and the cross-linked structure is generated. After the chemical reaction, the heat of formation Hf is considered by increasing the kinetic energy of the reacted part. The heat of formation increases the local temperature and accelerates the following reactions. The activation energy and heat of formation are summarized in Table 1. Both the activation energy of TGDDM and 44DDS are slightly higher than those of DGEBA and 44DDS. Figure 4 shows the evolution of the cross-linked structures at 473 K. An isolated structure is indicated by a different color. It is clearly understood that the cross-linked structure increased with the conversion of the reaction. In case of DGEBA/44DDS, the percolated network emerged at 95% conversion. However, full percolation occurred at 65% conversion in the case of TGDDM/44DDS owing to the difference in the number of epoxy groups. From these simulations, we can understand how the cross-linking reaction proceeds. Figure 5 shows a change in the different amino groups as a function of conversion. The ratio of the secondary amino group increases to reach its maximum value 0.5 at about 50% conversion in the case of DGEBA/44DDS. This means that the rates of production and consumption of the secondary amino groups are equivalent, and this tendency agrees with the experimental observation via near infrared spectroscopy (Min, 1993). The ratio of the secondary amino groups of TGDDM/44DDS, however, shows asymmetric change for conversion reaching its maximum value 0.35 at about 40% conversion. It is supposed that the second reaction in Equation 2 is slower than the first one in Equation 1. This difference is also caused by the number of epoxy groups and the chemical structure and not by the energetic barrier because the 1st and 2nd activation energy have the same magnitude. After cooling the cross-linked structure to 298 K, for each direction (x, y, z), a number of strains (0–0.1) are applied by MD simulation, resulting in a strained structure. Young’s modulus is estimated by averaging the stress–strain relationship for all directions, as shown in Table 2. The value of Young’s modulus tends to increase as the conversion increases, and this tendency is more noticeable and Young’s modulus is much higher at a high conversion for TGDDM/44DDS. Thermal properties, i.e., the coefficient of thermal expansion and the glass transition temperature, can be obtained by monitoring the relationship between the volume and temperature of the system. Next, the interfacial adhesion strength between a filler and epoxy resin is calculated. Here, we consider a copper substrate with (111) free surface as a filler, because the surface of carbon fiber is not fully clarified and a previous study has investigated a copper surface by MD (Yang, 2010). If an atomistic structure of the carbon fiber surface is clearly understood, we can create its model and perform similar simulations. A mixture of epoxy and amine compounds is placed over the copper surface, and then, the curing reaction is simulated to obtain the initial structures for peel-off simulation. Periodic boundaries are considered in the system, and therefore, it is assumed to be a sandwich structure of epoxy resin between the upper and lower copper layers. When the copper layer is fixed, a small displacement is applied in perpendicular to the copper surface and the stress is monitored via MD simulation. Two distinctive snapshots are shown in Figure 6 for 65% and 95% conversion for TGDDM/44DDS. Note that strain 1 refers to 5.7 nm displacement. In case of 65% conversion, a void is initiated within the epoxy resin at strain 0.2, and the voids grew in size and several chain segments of the cross-linked network become stretched (Strain 0.6 and 1.0). 5 Even at strain 1.0, the epoxy resin adheres to the copper surfaces. In contrast, the epoxy resin is peeled off at the interface in case of 95% conversion. This may be caused by the high rigidity of epoxy resin. The corresponding stress–strain relationships are shown in Figure 7. At high conversion of 80% and 95%, the yield stress is clearly recognized owing to peel-off at the interface.

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